Quantum Duality for Group Representations and Applications to Quantum Money and Fire

Core Concepts

This paper presents a novel quantum duality principle that equates the complexity of implementing a group representation to the complexity of performing a Fourier subspace extraction from its invariant subspaces, with significant implications for constructing secure quantum money and fire protocols.

Abstract

• Bibliographic Information: Bostanci, J., Nehoran, B., & Zhandry, M. (2024). A General Quantum Duality for Representations of Groups with Applications to Quantum Money, Lightning, and Fire. arXiv preprint arXiv:2411.00529.
• Research Objective: This paper aims to generalize the Aaronson, Atia, and Susskind (AAS) duality principle to a broader context of quantum states and multidimensional subspaces, and apply this generalized duality to advance the field of quantum cryptography, specifically in constructing secure quantum money and fire protocols.
• Methodology: The authors develop a theoretical framework based on group representation theory and quantum information processing. They introduce the concept of "Fourier subspace extraction" and establish a duality between implementing group representations and performing this extraction. This duality is then leveraged to design and analyze new cryptographic protocols.
• Key Findings:
  • The paper introduces a generalized duality principle for quantum computation, stating that the ability to implement a unitary representation of a group is computationally equivalent to performing a Fourier subspace extraction from its invariant subspaces.
  • This duality enables the construction of the first quantum lightning scheme with a rigorous security proof based on a plausible cryptographic assumption – the pre-action security of cryptographic group actions.
  • The authors also present a novel quantum money and lightning construction from one-way homomorphisms, demonstrating the equivalence of four distinct security notions.
  • Finally, the paper proposes the first candidate construction of quantum fire in the plain model, based on one-way group homomorphisms.
• Main Conclusions: The generalized quantum duality principle provides a powerful tool for understanding and manipulating quantum information, with direct applications in quantum cryptography. The proposed constructions for quantum money, lightning, and fire offer promising directions for developing secure quantum protocols based on concrete cryptographic assumptions.
• Significance: This research significantly advances the theoretical understanding of quantum computation and its applications in cryptography. The introduction of the generalized duality principle and its application to quantum money and fire constructions pave the way for further exploration and development of secure quantum protocols.
• Limitations and Future Research: While the paper provides a plausible construction for quantum fire, its security relies on the unproven security of one-way group homomorphisms. Further research is needed to investigate the existence and properties of such homomorphisms and to explore alternative constructions of quantum fire. Additionally, exploring other applications of the generalized duality principle in quantum computing and cryptography remains a promising avenue for future research.

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A General Quantum Duality for Representations of Groups with Applications to Quantum Money, Lightning, and Fire

by John Bostanc... at arxiv.org 11-04-2024

https://arxiv.org/pdf/2411.00529.pdf

Deeper Inquiries

How might this generalized duality principle be applied to other areas of quantum information science, such as quantum communication or quantum error correction?

The generalized duality principle presented in the paper, which links the ability to implement a unitary representation of a group to the ability to perform Fourier subspace extraction, has the potential to be a versatile tool with applications beyond quantum money and cryptography. Here are some potential avenues for its application in quantum communication and error correction:
Quantum Communication:

Secret Sharing and Quantum Key Distribution: The archetype states arising from Fourier subspace extraction could be used to design novel quantum secret sharing schemes. The ability to distinguish between different subspaces, as illustrated with the swap test example, could be leveraged to share secrets among multiple parties such that only authorized subsets can reconstruct them. This principle could also be extended to develop new quantum key distribution protocols, where the security relies on the hardness of distinguishing between different group representations.
Quantum Communication Complexity: The duality principle could lead to new lower bounds in quantum communication complexity. By relating the communication complexity of a function to the representation complexity of a related group, one might be able to leverage the duality to establish new bounds on the amount of quantum communication required to compute certain functions.
Blind Quantum Computation: The ability to perform Fourier subspace extraction could be useful in blind quantum computation protocols. A client could encode their computation as a representation of a group, and a server could perform the computation blindly by implementing the representation. The client could then use Fourier subspace extraction to recover the result of the computation without revealing the computation itself to the server.
Quantum Error Correction:

Designing New Quantum Codes: The invariant subspaces of group representations are inherently robust to errors that respect the group symmetry. This property could be exploited to design new quantum error-correcting codes based on non-Abelian groups. The duality principle could then be used to analyze the efficiency of encoding and decoding procedures for these codes.
Fault-Tolerant Quantum Computation: The duality could be useful in developing new fault-tolerant quantum computation schemes. By designing quantum gates that implement representations of certain groups, one might be able to leverage the duality to develop more efficient error correction and fault-tolerance mechanisms.
These are just a few potential directions, and further exploration of the generalized duality principle is likely to uncover even more applications in these and other areas of quantum information science.

Could there be alternative cryptographic assumptions, perhaps related to other aspects of non-Abelian group theory, that could be used to construct secure quantum money and fire?

Yes, alternative cryptographic assumptions related to non-Abelian group theory could potentially lead to secure quantum money and fire constructions. Here are a few possibilities:

Hardness of Subgroup Membership Problems: Many cryptographic assumptions rely on the hardness of deciding whether an element belongs to a particular subgroup. For instance, the Decisional Diffie-Hellman (DDH) assumption can be formulated as a subgroup membership problem. Generalizing this to non-Abelian groups, one could assume the hardness of distinguishing between a random element of a group and a random element from a specific subgroup. This could potentially be used to construct quantum money where banknotes are associated with hidden subgroups.
Braid Group Problems: Braid groups are non-Abelian groups with connections to knot theory and topology. Certain problems in braid groups, such as the conjugacy problem or the word problem, are believed to be computationally hard even for quantum computers. These hardness assumptions could potentially be used to construct quantum money or fire schemes where the security relies on the difficulty of solving these problems.
Lattice-Based Cryptography and Non-Abelian Groups: Lattice-based cryptography is a promising area for post-quantum cryptography. Recent work has explored connections between lattices and non-Abelian groups. It might be possible to leverage these connections to develop new cryptographic assumptions based on the hardness of lattice problems in a non-Abelian setting, which could then be used to construct quantum money and fire.
Quantum Hardness of Representation Theory Problems: Beyond classical assumptions, one could explore assumptions based on the quantum hardness of certain representation theory problems. For example, the problem of decomposing a given representation into its irreducible components could be computationally hard for certain groups, even for quantum computers. This hardness assumption could potentially be used to construct quantum money or fire schemes.
These are just a few examples, and further research into the computational hardness of various non-Abelian group theory problems could uncover even more promising candidates for cryptographic assumptions. The key is to identify problems that are believed to be hard even for quantum computers and then find ways to leverage those hardness assumptions to build secure cryptographic primitives.

What are the potential implications of developing practical and scalable quantum fire protocols for real-world applications like secure communication or data storage?

Developing practical and scalable quantum fire protocols, which enable the efficient cloning of quantum states that are hard to telegraph, could have significant implications for real-world applications like secure communication and data storage:
Secure Communication:

Unforgeable Quantum Tokens: Quantum fire states could be used to create unforgeable quantum tokens for authentication or access control. These tokens could be easily copied and distributed within a trusted network, but any attempt to transmit them outside the network would be detectable due to the difficulty of telegraphing.
Quantum Broadcast Authentication: Quantum fire could enable secure broadcast authentication schemes. A sender could distribute a quantum fire state as a signature, and all legitimate receivers could verify the signature by cloning the state. However, an adversary would be unable to forge the signature or rebroadcast it to unauthorized parties without detection.
Device-Independent Quantum Key Distribution: Quantum fire could potentially be used to develop new device-independent quantum key distribution protocols. These protocols would allow secure key distribution even if the quantum devices used are untrusted, as the security would rely on the inherent properties of the quantum fire states rather than the trustworthiness of the devices.
Data Storage:

Copy-Protection for Quantum Data: Quantum fire could provide a new approach to copy-protection for sensitive quantum data. By encoding the data into quantum fire states, one could allow authorized users to make copies while preventing unauthorized duplication or leakage of the data.
Secure Quantum Cloud Storage: Quantum fire could enhance the security of quantum cloud storage. Users could store their data encrypted with keys embedded in quantum fire states. This would allow them to access and process their data on the cloud while preventing the cloud provider from gaining access to the data or making unauthorized copies.
Other Implications:

New Cryptographic Primitives: The development of practical quantum fire protocols could lead to the discovery and development of new cryptographic primitives and functionalities beyond those mentioned above.
Fundamental Insights into Quantum Information: Quantum fire touches upon fundamental questions about the nature of quantum information, particularly the relationship between clonability and telegraphability. Further research in this area could lead to deeper insights into the nature of quantum information and its limitations.
However, realizing these potential benefits requires overcoming significant challenges. Practical quantum fire protocols need to be efficient, scalable, and robust to noise and errors. Additionally, the security assumptions underlying these protocols need to be carefully analyzed and validated. While significant hurdles remain, the potential benefits of quantum fire make it a promising area of research with the potential to revolutionize secure communication and data storage in the quantum era.